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  "body": "\n## Statement\n\nLet $x, y, z \\ge 0$ and $t \u003e 0$ be real numbers.\nThen\n$$x^t (x - y)(x - z) + y^t (y - z)(y - x) + z^t (z - x)(z - y) \\ge 0.$$\nEquality holds if and only if $x = y = z$, or two of $x, y, z$ are equal and the third is $0$.\n\n## Remarks\n\nProof idea: the expression is symmetric, so assume $x \\ge y \\ge z$.\nGroup the first two terms as $(x - y)\\big(x^t (x - z) - y^t (y - z)\\big)$, which is $\\ge 0$ because $x^t \\ge y^t \\ge 0$ and $x - z \\ge y - z \\ge 0$.\nThe last term $z^t (x - z)(y - z)$ is also $\\ge 0$.\n\nExample: for $t = 1$, expanding gives\n$$x^3 + y^3 + z^3 + 3xyz \\ge xy(x + y) + yz(y + z) + zx(z + x).$$\nIn terms of the [[elementary symmetric polynomials|0ZMY8441GW]] $p = x + y + z$, $q = xy + yz + zx$ and $r = xyz$, this reads $p^3 - 4pq + 9r \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nSchur's inequality is a sum of three terms, one for each of three nonnegative numbers.\nSome of the terms can be negative, but the total never is.\nThe case $t = 1$ is a cubic inequality that is often the missing step in symmetric problems.\n\n### L2 human\n\nSchur's inequality is a degree $t + 2$ symmetric inequality that does not follow from [[Muirhead's inequality|PBPFVY20WT]] alone, because it has negative coefficients.\nIt is a standard companion of Muirhead's inequality in proofs of symmetric inequalities in three variables.\nIts equality cases, which include $(1, 1, 0)$, show that it is sharp at the boundary as well as at the centre.\n",
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